Subgroups — #17053170

Order 4352.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531741	C₂	1	2	yes	maximal
2	#14529554	C₂	1	4	yes	maximal
3	#14529548	C₂	1	2	yes	maximal
4	#14528948	C₂	1	2	yes	maximal
5	#14528944	C₂	1	1	yes	maximal, characteristic
6	C₄×#10587489	C₂	1	2	yes	maximal
7	C₄×#10587489	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587489	C₂²	1	2	yes	—
10	C₂×#10587489	C₂²	1	1	yes	characteristic
11	C₂×#10587489	C₂²	1	2	yes	—
12	C₂×#10587489	C₂²	1	1	yes	characteristic
13	C₂×#10587488	C₂²	1	4	yes	—
14	#11762716	—	2	2	no	—
15	#11762695	—	2	4	no	—
16	#11762679	C₂²	1	2	yes	—
17	#11762675	C₂²	1	4	yes	—
18	C₄⋊C₄×Dic₁₇	—	2	2	no	—
19	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₁₇	C₂²	1	4	yes	—
21	#11762572	C₂²	1	4	yes	—
22	#11762572	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₁₇	C₄	1	8	yes	complemented
25	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₁₇	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₇	—	2	2	no	—
40	C₂×C₄⋊Dic₁₇	—	2	2	no	—
41	C₂×C₄⋊Dic₁₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₁₇	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₁₇	C₂³	1	2	yes	—
44	C₂×C₄×Dic₁₇	—	2	2	no	—
45	C₂×C₄×Dic₁₇	—	2	4	no	—
46	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₁₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₁₇	C₂×C₄	1	8	yes	—
50	#10587502	—	2	2	no	—
51	#10587502	—	2	4	no	—
52	#10587502	—	2	4	no	—
53	#10587502	—	2	4	no	—
54	#10587489	—	2	4	no	—
55	#10587489	—	2	2	no	—
56	#10587489	—	4	2	no	—
57	#10587489	Q₈	1	4	yes	complemented
58	#10587488	—	4	4	no	—
59	C₂×Q₈×C₁₇	—	2	2	no	—
60	C₂×Q₈×C₁₇	—	2	2	no	—
61	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₁₇	—	2	2	no	—
68	C₂²×Dic₁₇	—	2	2	no	—
69	C₂²×Dic₁₇	—	2	4	no	—
70	C₂×C₈×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	—	2	1	no	—
72	C₂×C₈×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	—	2	1	no	—
74	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₇	#41	1	2	yes	—
79	C₄⋊C₄×C₁₇	C₄⋊C₄	1	8	yes	—
80	C₄⋊C₄×C₁₇	C₄⋊C₄	1	4	yes	—
81	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
82	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₇	C₂×Q₈	1	4	yes	—
84	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₁₇	—	2	2	no	—
86	C₄⋊Dic₁₇	—	2	2	no	—
87	C₄⋊Dic₁₇	—	2	4	no	—
88	C₄⋊Dic₁₇	—	2	2	no	—
89	C₄⋊Dic₁₇	—	2	4	no	—
90	C₄⋊Dic₁₇	—	4	4	no	—
91	C₄⋊Dic₁₇	—	8	2	no	—
92	C₄⋊Dic₁₇	C₂×Q₈	1	4	yes	—
93	C₄×Dic₁₇	—	4	4	no	—
94	C₄×Dic₁₇	—	8	4	no	—
95	#22727	—	17	1	no	maximal, Sylow
96	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₁₇	—	2	8	no	—
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₇	#135	1	2	yes	—
109	C₂×C₄×C₁₇	#135	1	2	yes	—
110	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×Dic₁₇	—	2	4	no	—
117	C₂×Dic₁₇	—	4	2	no	—
118	C₂×Dic₁₇	—	4	2	no	—
119	C₂×Dic₁₇	—	4	2	no	—
120	C₂×Dic₁₇	—	4	4	no	—
121	C₂×Dic₁₇	—	4	2	no	—
122	C₂×Dic₁₇	—	4	4	no	—
123	C₂×Dic₁₇	—	4	4	no	—
124	C₂×Dic₁₇	—	4	2	no	—
125	C₈×C₁₇	—	2	4	no	—
126	C₈×C₁₇	—	2	2	no	—
127	C₈×C₁₇	C₄×Q₈	1	4	yes	—
128	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
129	#2355	—	17	2	no	—
130	#1949	—	17	4	no	—
131	#1948	—	17	2	no	—
132	#1846	—	17	2	no	—
133	#1845	—	17	1	no	—
134	C₄×#107	—	17	2	no	—
135	C₄×#107	—	17	1	no	—
136	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₇	D₄×Q₈	1	1	yes	characteristic
138	C₂²×C₁₇	#550	1	1	yes	characteristic
139	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
140	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
141	C₂²×C₁₇	#576	1	1	yes	characteristic
142	C₂²×C₁₇	#578	1	1	yes	characteristic
143	C₄×C₁₇	—	2	4	no	—
144	C₄×C₁₇	—	2	4	no	—
145	C₄×C₁₇	—	2	2	no	—
146	C₄×C₁₇	#427	1	4	yes	—
147	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₁₇	#520	1	1	yes	characteristic
149	C₄×C₁₇	#557	1	1	yes	characteristic
150	C₄×C₁₇	Q₈²	1	1	yes	characteristic
151	Dic₁₇	—	4	4	no	—
152	Dic₁₇	—	8	2	no	—
153	Dic₁₇	—	8	4	no	—
154	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
155	C₈×Q₈	Dic₁₇	1	8	yes	complemented
156	C₂×#107	—	17	1	no	—
157	C₂×#107	—	17	2	no	—
158	C₂×#107	—	17	1	no	—
159	C₂×#107	—	17	2	no	—
160	C₂×#106	—	17	4	no	—
161	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
163	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
165	#391	—	34	2	no	—
166	#387	—	34	4	no	—
167	#384	—	17	2	no	—
168	#382	—	17	4	no	—
169	C₄×C₄⋊C₄	—	17	2	no	—
170	C₄×C₄⋊C₄	—	17	4	no	—
171	C₄×C₄⋊C₄	—	34	2	no	—
172	#340	—	17	4	no	—
173	#340	—	17	2	no	—
174	C₂×C₁₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₁₇	C₂×#427	1	1	yes	characteristic
176	C₂×C₁₇	#2913	1	1	yes	characteristic
177	C₂×C₁₇	#3380	1	1	yes	characteristic
178	C₂×C₁₇	#3383	1	1	yes	characteristic
179	C₂×C₁₇	#3393	1	1	yes	characteristic
180	C₂×C₁₇	#3396	1	1	yes	characteristic
181	C₂²×Q₈	D₆₈	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
183	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
185	C₄×Q₈	Dic₃₄	1	4	yes	—
186	C₄×Q₈	C₂×Dic₁₇	1	4	yes	—
187	C₂×C₄⋊C₄	—	17	2	no	—
188	C₂×C₄⋊C₄	—	17	2	no	—
189	C₂×C₄⋊C₄	—	17	4	no	—
190	C₂×C₄⋊C₄	—	34	2	no	—
191	C₂×C₄⋊C₄	—	34	2	no	—
192	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
193	C₂×C₄⋊C₄	D₆₈	1	2	yes	—
194	C₂×C₄²	—	34	2	no	—
195	C₂×C₄²	—	34	4	no	—
196	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
197	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
198	#107	—	17	4	no	—
199	#107	—	34	4	no	—
200	#107	—	34	2	no	—
201	#107	—	68	2	no	—
202	#106	—	68	4	no	—
203	C₄⋊C₈	C₂×Dic₁₇	1	4	yes	—
204	C₄⋊C₈	C₂×Dic₁₇	1	8	yes	—
205	C₄×C₈	C₂×Dic₁₇	1	4	yes	—
206	C₄×C₈	C₂×Dic₁₇	1	8	yes	—
207	#95	—	34	2	no	—
208	#95	—	34	4	no	—
209	#95	—	34	4	no	—
210	#95	—	34	4	no	—
211	C₁₇	#22727	1	1	yes	characteristic, complemented, Sylow
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
215	C₂²×C₄	—	34	2	no	—
216	C₂²×C₄	—	34	2	no	—
217	C₂²×C₄	—	34	4	no	—
218	C₂²×C₄	C₂×D₆₈	1	2	yes	—
219	C₂²×C₄	C₂×D₆₈	1	2	yes	—
220	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
222	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	Q₈×D₁₇	1	2	yes	—
228	C₂×C₈	#63256	1	2	yes	—
229	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
230	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
231	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
232	C₄⋊C₄	—	17	4	no	—
233	C₄⋊C₄	—	34	2	no	—
234	C₄⋊C₄	—	34	2	no	—
235	C₄⋊C₄	—	34	4	no	—
236	C₄⋊C₄	—	34	2	no	—
237	C₄⋊C₄	—	34	4	no	—
238	C₄⋊C₄	—	68	4	no	—
239	C₄⋊C₄	—	136	2	no	—
240	C₄⋊C₄	C₄⋊Dic₁₇	1	8	yes	—
241	C₄⋊C₄	C₄⋊Dic₁₇	1	4	yes	—
242	C₄²	—	68	4	no	—
243	C₄²	—	136	4	no	—
244	C₄²	C₂×Dic₃₄	1	4	yes	—
245	C₄²	C₂×Dic₃₄	1	2	yes	—
246	C₄²	C₂²×Dic₁₇	1	4	yes	—
247	C₄²	C₂²×Dic₁₇	1	2	yes	—
248	C₂³	C₂²×D₆₈	1	1	yes	characteristic
249	Q₈	—	2	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	34	4	no	—
257	C₂×C₄	—	68	2	no	—
258	C₂×C₄	—	68	2	no	—
259	C₂×C₄	—	68	2	no	—
260	C₂×C₄	—	68	4	no	—
261	C₂×C₄	—	68	2	no	—
262	C₂×C₄	—	68	4	no	—
263	C₂×C₄	—	68	4	no	—
264	C₂×C₄	—	68	2	no	—
265	C₂×C₄	#10587590	1	2	yes	—
266	C₂×C₄	#10587590	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₁₇	1	4	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊Dic₁₇	1	4	yes	—
270	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
271	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#63256	1	1	yes	characteristic
274	C₂×C₄	#10587696	1	1	yes	characteristic
275	C₂×C₄	#10587697	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	2	2	no	—
279	C₈	Q₈×Dic₁₇	1	4	yes	—
280	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
281	C₂²	Q₈×D₆₈	1	1	yes	characteristic
282	C₂²	#11763481	1	1	yes	characteristic
283	C₂²	C₂×#10587590	1	1	yes	characteristic
284	C₂²	C₂×#10587590	1	1	yes	characteristic
285	C₂²	#11763614	1	1	yes	characteristic
286	C₂²	#11763626	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	68	4	no	—
291	C₄	—	136	2	no	—
292	C₄	—	136	4	no	—
293	C₄	#11762847	1	4	yes	—
294	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
295	C₄	#11763292	1	1	yes	characteristic
296	C₄	#11763538	1	1	yes	characteristic
297	C₄	Q₈×Dic₃₄	1	1	yes	characteristic
298	C₂	Q₈×C₄⋊Dic₁₇	1	1	yes	characteristic
299	C₂	C₂×#11762847	1	1	yes	characteristic
300	C₂	#14537384	1	1	yes	characteristic
301	C₂	#14541652	1	1	yes	characteristic
302	C₂	#14541676	1	1	yes	characteristic
303	C₂	#14541773	1	1	yes	characteristic
304	C₂	#14541797	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×D₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	#22727
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
24. C₈×Q₈×C₁₇	C₄	semidirect
57. #10587489	Q₈	semidirect
155. C₈×Q₈	Dic₁₇	semidirect
211. C₁₇	#22727	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14531741	2
#14529554	2
#14529548	2
#14528948	2
#14528944	2
C₄×#10587489	2
C₂×C₈×Q₈×C₁₇	2
#22727	17